The humidity level in a bath is usually pretty high. The thinner the wood you use the more likely you would have some cup warpage. It might be better if you glued the 5/16" wood to some other solid wood.
I would recommend what ever finish you use you finish both sides to keep it as stable as possible. I don't have any experience with the Rubio finish but I've heard bad things. If it were me I would use a marine grade spar varnish. Epifanes is the best but a little expensive.